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The Leray-Schauder Principle and Applications

In: Methods in Nonlinear Integral Equations

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  • Radu Precup

    (Babeş-Bolyai University, Department of Applied Mathematics)

Abstract

In applications one of the drawbacks of Schauder’s fixed point theorem is the invariance condition T (D) ⊂ D which has to be guaranteed for a bounded closed convex subset D of a Banach space. The Leray-Schauder principle [32] makes it possible to avoid such a condition and requires instead that a ‘boundary condition’ is satisfied. In this chapter we shall prove the Leray-Schauder principle and we shall apply it in order to obtain existence results for continuous solutions of integral equations. In particular, we give results on the existence of continuous solutions of initial value and two-point boundary value problems for nonlinear ordinary differential equations in R n . The results will be better than those established by means of Schauder’s theorem.

Suggested Citation

  • Radu Precup, 2002. "The Leray-Schauder Principle and Applications," Springer Books, in: Methods in Nonlinear Integral Equations, chapter 0, pages 43-60, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9986-3_5
    DOI: 10.1007/978-94-015-9986-3_5
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